A 1-Dimensional Diagram For Extensive Rent With Markup Pricing

Figure 1: Extensive Rent Example As Relative Market Power Varies Between Industry And Agriculture

This post is an elaboration on this past post.

The analysis of the choice of technique varies with perturbations of relative markups in industry and agriculture. Figure 1 depicts this variation, while Figure 2 is an enlargement of the lower range of the relative markup in industry. The heavy solid lines in Figure 1, other than the horizontal line at the top, are switch points on the inner frontier of the wage curves. They divide the space into areas in which the cost-minimizing technique is labeled in bold. The dashed lines are intersections on the outer frontier of the wage curves. The order of fertility varies across dashed lines. The dotted lines are intersections of rent curves. The order of rentability varies across dotted lines.

Figure 2: Extensive Rent Example As Relative Market Power Varies Between Industry And Agriculture (Enlarged)

Fluke cases partition the graph (Table 1), as shown by the thin vertical lines and the numbering of the resulting regions. To call the intersections of wage curves on the outer frontier 'switch points' is an abuse of language. Quantity flows and the technique do not vary around these intersections. The same types of land are fully farmed, and the type of land that is only partially farmed is farmed to the same extent. The same quantity of iron is produced. Nevertheless, the order of efficiency varies around such intersections. Likewise, to call the intersections of rent curves ‘'switch points' is a similar abuse of language. The order of rentability varies around these intersections.

Table 1: Fluke Switch Points

Regions	Description
Between 1 and 2	Rent curves for Type 1 and 3 lands tangent at switch point.
Between 2 and 3	Switch point for rent curves for Type 1 and 2 lands at wage of zero.
Between 3 and 4	Switch point for wage curves for Beta and Gamma at wage of zero.
Between 4 and 5	Switch point for wage curves for Alpha and Beta at wage of zero.
Between 5 and 6	Three wage curves intersect at single switch point.
Between 6 and 7	Wage curves for Alpha and Beta tangent at switch point.
Between 7 and 8	Switch point for rent curves for Type 1 and 2 lands at wage of zero.

The analysis of the choice of technique is qualitatively invariant in each numbered region. The number and sequence of intersections of wage curves and of rent curves do not vary within a numbered region. Table 2 lists the cost-minimizing technique along the wage frontier, from a wage of zero to the maximum wage, in each numbered region. Variations in the order of fertility and in the order of rentability are also indicated.

Table 2: Variations in the Cost-Minimizing Technique

Region	Range	Technique	Order of Fertility	Order of Rentability
1	0 ≤ w ≤ w1	Beta	Type 1, 3, 2	Type 3, 1, 2
	w1 ≤ w ≤ w2		Type 3, 1, 2
	w2 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
2	0 ≤ w ≤ w1	Beta	Type 1, 3, 2	Type 3, 1, 2
	w1 ≤ w ≤ w2			Type 1, 3, 2
	w2 ≤ w ≤ w3			Type 3, 1, 2
	w3 ≤ w ≤ w4		Type 3, 1, 2
	w4 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
3	0 ≤ w ≤ w1	Beta	Type 1, 3, 2	Type 1, 3, 2
	w1 ≤ w ≤ w2			Type 3, 1, 2
	w2 ≤ w ≤ w3		Type 3, 1, 2
	w3 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
4	0 ≤ w ≤ w1	Gamma	Type 1, 2, 3	Type 1, 2, 3
	w1 ≤ w ≤ w2	Beta	Type 1, 3, 2	Type 1, 3, 2
	w2 ≤ w ≤ w3			Type 3, 1, 2
	w3 ≤ w ≤ w4		Type 3, 1, 2
	w4 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
5	0 ≤ w ≤ w1	Gamma	Type 2, 1, 3	Type 1, 2, 3
	w1 ≤ w ≤ w2		Type 1, 2, 3
	w2 ≤ w ≤ w3	Beta	Type 1, 3, 2	Type 1, 3, 2
	w3 ≤ w ≤ w4			Type 3, 1, 2
	w4 ≤ w ≤ w5		Type 3, 1, 2
	w5 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
6	0 ≤ w ≤ w1	Gamma	Type 2, 1, 3	Type 1, 2, 3
	w1 ≤ w ≤ w4		Type 1, 2, 3
	w2 ≤ w ≤ w3		Type 2, 1, 3
	w3 ≤ w ≤ w4			Type 2, 1, 3
	w4 ≤ w ≤ w5	Alpha	Type 2, 3, 1	Type 2, 3, 1
	w5 ≤ w ≤ w6			Type 3, 2, 1
	w6 ≤ w ≤ wα, max		Type 3, 2, 1
7	0 ≤ w ≤ w1	Gamma	Type 2, 1, 3	Type 1, 2, 3
	w1 ≤ w ≤ w2			Type 2, 1, 3
	w2 ≤ w ≤ w3	Alpha	Type 2, 3, 1	Type 2, 3, 1
	w3 ≤ w ≤ w4			Type 3, 2, 1
	w4 ≤ w ≤ wα, max		Type 3, 2, 1
8	0 ≤ w ≤ w1	Gamma	Type 2, 1, 3	Type 2, 1, 3
	w1 ≤ w ≤ w2	Alpha	Type 2, 3, 1	Type 2, 3, 1
	w2 ≤ w ≤ w3			Type 3, 2, 1
	w3 ≤ w ≤ wα, max		Type 3, 2, 1

In region 1 at the extreme left in Figure 1, agriculture has overwhelming market power, as compared to industry. Owners of Type 2 land do not obtain a rent except when workers make a high wage. Owners of Type 3 land obtain the highest rent per acre, whatever the order of efficiency. In region 8 at the extreme right, owners of Type 3 land do not obtain a rent when workers make a low wage. In this range, industry has much more market power than agriculture. The possibilities are complicated in between these extremes.

Region 5 includes the competitive case, in which s₁ = 1/2. As previously noted, this region exhibits a reswitching of the order of efficiency. Region 2 exhibits a reswitching of the order of rentability. I find worth emphasizing the ranges of the wage in each region in which the orders of efficiency and rentability differ. Such a range exists for each numbered region. Landlords with less fertile land can receive more rent per acre than those with more fertile land. Under capitalism, you are not rewarded, in general, for your contributions or for the contributions of the resources that you own to production. This disconnect applies both to competitive and non-competitive markets.

The analysis of the choice of technique with extensive rent is more complicated than such analysis in a model with only circulating capital:

"The complexity of the outcomes with the potential existence of conflict or concordance among the three major economic categories (earners of wages, profits, and rents) profoundly modifies the traditional analysis of profits and wages." -- Alberto Quadrio Curzio and Fausta Pellizzari (2010).

The quantity and price systems are interconnected. Assumptions on the level of net output are required to determine which techniques are feasible. All three types of land need to be farmed, at least partially, to satisfy the requirements for use in the numeric example.

Introducing relative market power among industries further complicates the analysis of the choice of technique. Extensive rent and which land is marginal, that is, receives no rent, depend on the wage and the relative market power of industry and agriculture. Changes in market power can have similar qualitative effects, such as the introduction of the reswitching of the order of rentability, as structural economic dynamics resulting from technical innovation.